J . Phys Chem. 1988, 92, 1752-1755
1752
Transport Coefficients of Diatomic Gases: Internal-State Analysis for Rotational and Vibrational Degrees of Freedom Carl Nyeland* and Gert Due Billing University of Copenhagen, Chemistry Laboratory I l l , H . C . 0rsted Institute, DK-2100 Copenhagen 0 , Denmark, and Unioersity of Copenhagen, Department of Chemistry, Panum Institute, D K - 2200 Copenhagen N , Denmark (Receiued: July 13, 1987; In Final Form: October 7, 1987)
Rate coefficients for relaxation and transport (thermal conductivity, viscosity, self-diffusion) have been calculated for diatomic nitrogen in the low-pressure limit of bimolecular collisions. Classical dynamics have been used in Monte Carlo calculations of rotating, nonvibrating, and vibrating molecules, taking inelasticity fully into account. Site-site potentials of Berns and van der Avoird and of Ling and Rigby have been considered. From the collisional results obtained, one sees that the first-order treatment of the Wang Chang-Uhlenbeck theory is useful for calculations of transport coefficients for temperatures where the rigid rotor model is appropriate. The results at higher temperatures indicate that inclusion of higher order terms in the Wang Chang-Uhlenbeck theory or eventually a quantum mechanical treatment of vibration is necessary.
1. Introduction The theory of polyatomic gases was brought into a very useful form by Wang Chang and Uhlenbeck' in a report written in 1951. Prior to that the most extended treatment of thermal conductivity in a polyatomic gas had been given by Hirschfelder2 who obtained the expression = %(o/m)c,, + (pD/m)cint
The present paper is arranged in the following way. In section 2 the general results from the theory of Wang Chang and Uhlenbeck are recapitulated and the formulas for collision integrals are given. In section 3 the N2-N2 potentials considered are discussed. Calculations of collision integrals for rotating nonvibrating molecules, the rigid rotor model, are presented in section 4 and discussed for temperatures in the interval 100-600 K. In section 5 some results for collision integrals for rotating and vibrating molecules are given and discussed. In the final section a few conclusions obtained from the results are given, together with some comments upon the possibility of obtaining reasonable estimates of transport coefficient data from a primitive knowledge of intermolecular potentials.
(1)
where X is the thermal conductivity, 7 is the viscosity, m and p are the molecular mass and mass density, respectively, and c,, and c,, are the heat capacities for the translational and internal degrees of freedom, respectively. This result appeared from a model-type calculation, assuming that the contribution from the internal degrees of freedom can be added by using a local equilibrium approximation to the well-known result of Chapman and Enskog for monatomic gases. D in eq 1 was considered to be a diffusion constant for internal-state diffusion, neglecting inelastic collisions. Still earlier, Eucken3 had considered the form
= (o/m)(%ctr + Cint) (2) obtained by mean-free-path arguments. A very often used approximation for calculations of viscosities of polyatomic gases in recent years was suggested by Monchick and Mason.4a It is based on the Wang Chang-Uhlenbeck result' (see section 2), but assumes no rotation (or vibration) of the molecules during collisions. In the calculations presented in this paper (and in our earlier works5) we consider full inelastic collisions and calculate transport and relaxation coefficients following the detailed Wang Chang-Uhlenbeck treatment, but only in the limit of classical mechanics. Better methods would, of course, be considerations based on a quantum mechanical treatment of collision dynamics, particularly calculations following the full close coupling procedure. Such calculations were recently considered by Maitland et aL6 on the helium-nitrogen system. When results from more accurate methods appear, it will be possible to learn about the usefulness of classical mechanics in calculations of transport coefficients. By focusing on the Wang Chang-Uhlenbeck description, we have neglected contributions from the spin polarizations in the inhomogeneous gases considered: This means that no results for external field problems can be obtained. In ref 5a the errors expected in calculations following the Wang Chang-Uhlenbeck method for ordinary zero-field problems were briefly discussed. With a semiempirical argument, the expected error was shown to be less than 1 % for calculations of thermal conductivities and probably smaller for calculations of viscosities.' *To whom correspondence should be addressed at the H. C. 0rsted Institute.
0022-3654/88/2O92-l752$01.50/0
2. Wang Chnng-Uhlenbeck Theory The formal, bimolecular results for the transport coefficients obtained in the Wang Chang-Uhlenbeck theory have been discussed in ref 5a in the classical limit. Also the classical limit of the results for relaxation time and the pressure-broadening coefficient of the depolarized Rayleigh scattering (DPR) were given there. Here only the main points will be considered. The average symbol (I)) for the collision integrals is defined by
where the collision cross section
N d i ) + N2CI')
-
c;r
for the processes
N 2 ( k ) + NAO
(4)
with i denoting the set of quantum numbers (u, j , mi) is given by ~~
'
~~
~
( I ) Wang Chang, C. S.; Uhlenbeck, G . E. 'Transport Phenomena in Polyatomic Gases; Report No. CM-681; University of Michigan Engineering Research: Ann Arbor, MI, 1951. See also: Wang Chang, C. S.; Uhlenbeck, G. E.; de Boer, J . In Sfudies in Statistical Mechanics; de Boer, J . , Uhlenbeck, G. E., Eds.; North-Holland: Amsterdam, Netherlands, 1964; Vol. 2, Part C. (2) Hirschfelder. J. 0. J . Chem. Phys. 1957, 26, 282. See also: Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. 8. Molecular Theory of Gases and Liquids; Wiley: New York, 1964. (3) Eucken, A. Phys. Z 1913, 15. 324. See also: Ferziger, J. H.; Kaper, H. G. Mathematical Theory of Transport Processes in Gases: North-Holland: Amsterdam, Netherlands, 1972. (4) (a) Monchick, L.; Mason, E. A. J . Chem. Phys. 1961, 35, 1676. (b) Mason, E. A.: Monchick, I,. J . Chem. Phys. 1962, 36, 1622. (c) Monchick. L;Pereira, A. N . G.; Mason, E. A . J . Chem. Phys. 1965, 42, 3241. ( 5 ) (a) Nyeland, C.; Poulsen, L. L.; Billing, G. D. J . Phys. Chem. 1984, 88, 1216. (b) Ibid. 1984, 88, 5858. (6) Maitland, G. C.; Mustafa. M.; Wakeham, W. A,; McCourt. F. R. W. Mol. Phys. 1987, 61, 359. (7) The experimental results used in the argument were from the following: Hermans, L. J . F.; Koks, J . M . ; Hengeveld, A. F.; Knaap, H. F. P. Physica (Amsterdam) 1970, 50, 410. Burgmans, A . L. J.; van Ditzhuyzen. P G.: Knaap, H . F. P.; Beenakker, J . J. M . Z . Naturforsch., A : Phys.. Phys. C'hem, Kosmopkys. 1973. 28A. 835.
0 1988 American Chemical Society
Transport Coefficients of Diatomic Gases
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1753
ut’ dfl = 2*P$!(b)b db
(5)
P$$(b)is the transition probability and is given as a function of the impact parameter b. Furthermore, we have tj
= Ei/kT
(6)
yz = Y4mv2/kT QA
=
TABLE I: Collision Integrals for the Rigid Rotor Model (T = 100-600 K)a T, K eq 10 eq 11 eq 12 eq 13 eq 14 Potential Model A 100 200 300 400
(7)
Ce-+i
500
1
At = c k
+ tl -
ti
600
- tj
100 200 300 400
500
(10)
((Y2 sin2 6 ) )
+~(AE)~))
(12) (13)
- ? ) [ ( t i - t j ) + - (c,c - el)yy’ COS XI))
(14)
where 6 is defined as the angle by which the direction of the rotational angular momentum vector of one of the colliding molecules is changed due to the collision, x is the scattering angle in the relative system, y’ the reduced velocity after the collision, and z the average internal energy for a molecule. The following formal first-order results from the Wang Chang-Uhlenbeck method can be written by use of the five collision integrals 10-14. The bulk viscosity qv is given by 7,
= 7nk2Tcint/c>
where the relaxation time 7-1
600
(11)
((YZ - YY’ cos XI) (((ti
(15)
is given by 2nk =(l(A42)) 7
(16)
Cint
n is the density of molecules and c, the heat capacity at constant volume. The relaxation collision number { is defined as
4 P7 {= * T
where p is the pressure. The cross section for DPR scattering is given by
The shear viscosity q is given by 8 TJ-’ = -({y4 sin2 x - ‘/2(Ae)*sin2 x 5kT
0.317 0.284 0.267 0.255 0.249 0.243
0.362 0.331 0.314 0.302 0.294 0.287
Potential Model B 0.317 1.30 0.316 1.46 0.317 1.57 0.317 1.65 0.317 1.72 0.317 1.77
(9)
where v is the relative velocity and QA the partition function for internal states. The following five collision integrals are considered
((y4sin2 x - ) / ~ ( A Csin2 )~ x
0.379 0.336 0.313 0.298 0.287 0.278
+ Y3(At)2)) (19)
The coefficient of self-diffusion is given by 8nm D-1 = -((y2 - yy’ cos x)) 3kT The thermal conductivity is given by
1.31 1.38 1.42 1.45 1.48 1.52
0.556 0.606 0.638 0.661 0.680 0.697
0.692 0.716 0.731 0.742 0.750 0.766
0.558 0.637 0.688 0.726 0.758 0.784
0.657 0.664 0.668 0.670 0.672 0.674
“In units of lo-’o cm3 d. c’= cht/k. See ref 5a for further discussion of these formal results.
3. Intermolecular Potentials For the potential between two nitrogen molecules we have considered two examples of short-range site-site potentials combined with a dispersion term and a quadrupole-quadrupole interaction term. In potential model A we assumed the simple exponential site-site potential suggested by Berm and van der Avoird8 from their ab initio calculations
where A = 293.3 au, a = 2.1 36 au, and R is the distance between the atoms in different molecules. The c6 dispersion constant was set to 24.4 au8 and for the quadrupole moment Q the theoretical resultg -1.136 au was used. For vibrating molecules the quadrupole derivative also was taken into account. The theoretical value9 ( a Q / a r ) , = 0.632 au was used. In potential model B we assumed the extended potential sitesite potential suggested by Ling and Rigbylo
vsR= A ~ + R - @ R *
(25)
where A’= 22.568 au, a‘ = 1.186 au, and @ = 0.09 au. The same values as for model A were used for the dispersion constant c6 and the quadrupole moment Q, but the quadrupole derivative was set to the experimental value,” ( d Q / a r ) , = 0.933 au. The two different potentials yield slightly different collision integrals as will be discussed in the following sections. Also minor differences are seen when comparing results obtained here with results obtained earlier.sa No attempt was made to optimize a potential from the experimental information obtained from the different transport coefficients. Some further information about the N2-N2 potential has appeared recently: ref 12 of Hay et al., ref 13 of Biihm and Ahlrichs, and ref 14 of van der Avoird et al. They all have minor differences from the potential models A and B considered here and should be taken into account in a detailed study for the final determination of the potential between two nitrogen molecules.
4. Rigid Rotor Model Results
Here = pDint/v,where the coefficient of “internal diffusion” Dintis defined as
and the term Qf;;’) is given by
Ci”tfi[:il) k
= ( ( ( e i -?)[(ti - tj)y2 - ( e k - tl)yy’ cos
XI))
(23)
For the two potential models mentioned in section 3 , the five collision integrals from eq 10-14 were obtained following the Monte Carlo procedure discussed in ref 5a. For temperatures up ~~~
~~
~
(8) Berns, R. M.; van der Avoird, A. J . Chem. Phys. 1980, 72, 6107. (9) Billingsley, F. P., 11; Krauss, M. J . Chem. Phys. 1974, 60, 2767. (10) Ling, M. S . H.; Rigby, M. Mol. Phys. 1984, 51, 8 5 5 . (11) Reuter, D.; Jennings, D. E. J. Mol. Spectrosc. 1986, 115, 294. (12) Hay, P. J.; Pack, R. T.; Martin, R. L. J. Chem. Phys. 1984, 81, 1360. (13) Bohm, H.-J.; Ahlrichs, R. Mol. Phys. 1985, 55, 1159. (14) van der Avoird, A,; Wormer, P. E. S.;Jansen, A. P. J. J. Chem. Phys. 1986, 84, 1629.
Nyeland and Billing
1754 The Journal of Physical Chemistry. Vol. 92, No. 7, 1988 TABLE II: Rotational Relaxation Numbers and Transport Coefficients for the Rigid Rotor Model
r, K
f(WCU) > f(Eucken)
(27)
are most often fulfilled. Results obtained form the DPR cross section and particularly for potential model A agrees very well with an experimental value and also with the molecular collision calculations obtained by Turfa et aLZ0 At T = 300 K we obtained oDPR= 3 1.7 AZfor potential model A and 37.6 A2 for potential model B. At T = 293 K the results for the calculation given in ref 20 are 29.2 A2 and 3 1.6 A2 for slightly different potentials and 34.4 A2for the experimental value. A few qualitative features were furthermore obtained from the results: (20) Turfa, A. F.; Connor, J. N. L.; Thijsse, B. J.; Beenakker, J. J. M. Physira A (Amsterdam) 1985, 129A, 439.
Transport Coefficients of Diatomic Gases
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1755
TABLE I V Collision Integrals for Rotating and Vibrating Molecules for Potential Model B (T= 800-1600 K ) O ea 10 ea 11 ea 12 ea 13 ea 14 T = 800 K 0.317 1.87 0.677 R R ~ 0.276 0.828 00‘ Ole
0.228 0.297
0.296 0.322
1.70 1.55
0.716 0.724
0.663 0.918
RR
0.262 0.204 0.278
T = 1200 K 0.317 2.00 0.296 1.83 0.326 1.63
0.894 0.770 0.772
0.681 0.667 0.993
0.252 0.188 0.266
T = 1600 K 0.316 2.11 0.296 1.93 0.328 1.68
0.944 0.81 1 0.808
0.683 0.669 1.05
00 01
RR 00 01
cm3 s-I. bRR means results from the rigid rotor “ I n units of and 01 mean initial, semiclassical ‘states” of vibration for model. the colliding molecules.
The collision integrals for viscosity are around 5-6%, smaller when inelastic collisions are taken into account than those obtained from calculations based on the angular averaged potential. Similar but relatively larger contributions from the anisotropic part of the potentials have been observed in calculations of the second virial c~efficient.’~*~’ The quantity @ in the expression for the thermal conductivity, eq 21, is generally around 10% smaller than the value expected if the self-diffusion constant D is substituted for Dintin /3 as suggested by Mason and M ~ n c h i c k .A ~ similar reduction was also obtained by Assael et al.19following a semiempirical approach. The results for the relaxation number {are slightly larger than expected before detailed collision dynamics were possible. This compensates partly for the effect from the reduction in @ discussed above in the calculation of thermal conductivities.
5. Vibrating Rotor Model Results A series of molecular collisions have been considered where the vibrational degrees of freedom were allowed to interact fully with the other degrees of freedom following a classical mechanical description. In one series of calculations, the initial distribution of vibrational coordinates and momenta were selected by using the classical distribution function for rotating Morse oscillators corresponding to the vibrational energy Evib= ‘ / z h w for both colliding molecules.22 In another series, one of the molecules was ~ to the first given an energy equal to Evib= 3 / 2 hcorresponding excited state of the vibrator. Results for the collision integrals obtained for temperatures in the interval 800-1600 K are given in Table IV, together with rigid rotor results. As far as we know, the present calculation of transport coefficients is the first which has included the vibrational degree of freedom in a correct (albeit classical) dynamical treatment of the collision dynamics, also taking the scattering angles fully into account. The reason for including vibrational motions was the poor agreement obtained (21) Corbin, N.; Meath, W. J.; Allnatt, A. R. Mol. Phys. 1984, 53, 225. (22) Porter, R. N.; Raff, L. M. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Vol. B., p 1.
at high temperatures between our calculations and experimental results. However, to our surprise the results obtained for the collision integrals are smaller for vibrating rotors than they are for the rigid rotors. This corresponds to larger values of the transport coefficients (the experimental values are smaller). Two possible reasons for this deviation have to be considered: Errors due to a too easy vib-rot/trans energy exchange of part of the zero-point vibrational energy in classical mechanics might be important here. Similar unphysical vibrational exchange has recently been observed in polyatomic molecules.23 Only the first-order terms from the Wang Chang-Uhlenbeck expansion have been considered. It may be that higher order terms24 are necessary for the vibrating rotor model while the first-order results appear to be sufficient at lower temperatures, where the rigid rotor model seems to work fairly well. In both cases, calculation of transport coefficients at high temperatures appears to be very complicated and has to await further studies. In examining the results of Table IV, one sees that simple rules such as those suggested by Mason and M ~ n c h i c kbased ~ ~ upon the neglect of rot-vib energy exchange, for instance that 7vib
. ,
are not well fulfilled here. On the other hand, vib-vib energy exchange appears to be important, judging from the results for the vibrationel contributions to the internal diffusion constant. Roughly, one has that &(Ol) = Dint(OO)(l A)-’
-
+
where A 0.1-0.2 from the results in Table IV. Similar contributions to internal diffusion from rotational excitation transfer have been considered earlier.4s25
6. Discussion From the results obtained here for nitrogen and for the rigid rotor model, it can be argued that good agreement with experimental results for transport and relaxation coefficients for temperatures below 600 K can be obtained by minor adjustments of the intermolecular potential considered. The results for the vibrating rotor model seem to indicate an unphysical rot-vib/trans energy exchange in classical mechanics or that a higher order Wang Chang-Uhlenbeck treatment is necessary in calculation of transport and relaxation coefficients at temperatures above 600 K. Estimates for technical use can easily be made in the lowtemperature range based on any reasonable potential model eventually combined with an average value for the Eucken factor. Acknowledgment. This research was supported in part by the Danish Natural Science Research Council. We are grateful to Dr. F. M. Nicolaisen for bringing to our attention some recent experimental results for the quadrupole derivative. Registry No. N2, 7727-37-9. (23) Billing, G. D. Chem. Phys. 1986, 104, 19. (24) Maitland, G. C.; Mustafa, M.; Wakeham, W. A. J . Chem. Soc., Faraday Trans. 2 1983, 79, 1425. ( 2 5 ) Nyeland, C.; Mason, E. A.; Monchick, L. J . Chem. Phys. 1972,56, 6180.
